I have absolutely no experience with electronics but I am looking to create a custom pc lighting mod with the RGB strip HERE that will let me change the colour of the case lighting. I am basically looking for a programmable controller similar to the arduino but preferably cheaper. I don't really know what the controller needs in order for me to be able to program the strip and then disconnect it from the PC while still having it do whatever it was programmed to. I have plenty of experience programming in C and C++ so this side of things won't be a problem, all I want to know is what is the cheapest programmable controller I could use to do what I want. Any hints and tips would also be greatly appreciated

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You don't need a store bought Arduino for that. Just build a minimalist clone: Atmel 328p, 1x 0.1uF cap, 1x 16MHz resonator, and the LED strip. I have several of those strips running different light sequences as a Christmas display.

All you need to do is write your code for controlling the strip (I use the FastSPI library), upload it to the Atmel, connect the strip and off you go. As long as you can power the Atmel and light strip, you can put it anywhere. I use an old computer PSU for that.